固体地球物理学概论Snell定律课件.ppt

1、第七章第七章第七章第七章 地震波场在地球物理学中占有重要地位，当今在研究地球内部，地地震波场在地球物理学中占有重要地位，当今在研究地球内部，地震活动的机制、资源与能源的地震勘探以及海、陆工程建设中均主要依震活动的机制、资源与能源的地震勘探以及海、陆工程建设中均主要依赖于人工源或天然源激发的地震波场效应。赖于人工源或天然源激发的地震波场效应。震源震源(包括人工源或天然源包括人工源或天然源)激励出来的各种类型的地震波，在地球激励出来的各种类型的地震波，在地球内部各围层介质中或沿其表面传播，依据这些波动的内部各围层介质中或沿其表面传播，依据这些波动的走时走时，频率频率和和振幅振幅特性或波的特性或波的

2、频散频散，可以推测地球内部各圈层介质的速度分布和结构。，可以推测地球内部各圈层介质的速度分布和结构。根据地震台站纪录的地震事件，可推断震源的参数根据地震台站纪录的地震事件，可推断震源的参数(震源深度震源深度、震中震中位置、发震时刻、地震震级和震中距离位置、发震时刻、地震震级和震中距离等等)和震源机制，并进一步了解产和震源机制，并进一步了解产生这种机制的应力状态。如果发生的地震足够大，则地球作为一个整体生这种机制的应力状态。如果发生的地震足够大，则地球作为一个整体会激发出各种振型的振荡，并可通过它来探讨地球内部的性质。会激发出各种振型的振荡，并可通过它来探讨地球内部的性质。第七章第七章7.1 地

3、震与介质的弹性性质地震与介质的弹性性质 Earthquake sources Physically,earthquake sources are the abrupt release of the potential elastic energy stored in rocks over a period ranging from a few years to thousands of years.Only a small part of the energy converts heat to the surround rocks near the source,the most of ene

4、rgy is radiated away as elastic waves.In fact,rocks at an earthquake source generate plastic deformation but elastic deformation at the moment of earthquakes occurring.Up to now,no earthquake that locates deeper than 670 km has been observed.第七章第七章 Seismic waves Seismic waves are the elastic wave fr

5、om the source.They are of various types.While traveling through the earth,these waves are influenced by the properties of the media they pass.We are able to understand and analyze the influence by applying mathematical and physical methods.In addition,we can determine the earths structures:the crust

6、,the mantle,the outer core,the inner core,as well as the lateral changes near the surface.第七章第七章 Receiving The main task of receiving is to record the seismic waves at desired positions to study earthquakes.It includes the sampling technology and the receiving system consisting of seismographs(地震仪地震

7、仪),geophones(检波器检波器),and other instruments.The quality of recorded data is important to analysis and interpretation of the seismic waves and determination of earthquake sources.第七章第七章 At present the theory of plate tectonics is accepted by most of geoscientists,even though many of its details are st

8、ill unclear or controversial.We can use a simplified dynamic model to describe the movement of continent.When the material in the mantle is heated,it expands and becomes lighter.In spite of its high viscosity（粘性）（粘性）,it rises more or less vertically in some places,especially under the oceanic ridges

9、.With its losing pressure and heat during traveling upward,the material is forced to travel horizontally.They drag the lithosphere motion.The results of continent colliding form mountain chains(Himalayas),and the results of their separating form ocean rifts(East Africa).So major earthquakes often ca

10、use near their collided boundaries.In the region of oceanic ridges,where new lithosphere is growing,small earthquakes occur frequently.第七章第七章第七章第七章 (1)(1)形变形变 A material occurs deformation(形变）形变）under a force act on itself.If it recovers as the force disappears,it is called elastic material.The shap

11、e change is called as elastic deformation.Otherwise,it is called as non-elastic deformation.Whether elastic deformation occurs depends on the magnitude of acting force,the acting period as well as the surrounding temperature.For most materials in the earth,this elastic property only exists in a shor

12、t period.第七章第七章 (2)(2)应力应力 Stress tensor（张量）（张量）Definition:Here stress means a force acts on unit area of a body against the elastic deformation caused by the action of an external force.Describe any stress needs consider two factors,direction and outer normal(法向）of a face.We generally express it by

13、 pst.Here s means the direction of the force and t the outer normal direction on a face.In three dimension orthogonal coordinate system,we can define stress p as(pxx pxy pxz pyx pyy pyz pzx pzy pzz).第七章第七章 The stresses are symmetrical（对称的）,i.e.only six components of the stress tensor p are independe

14、nt because pxy=pyx,pyz=pzy,pzx=pxz For a cubic body in x-y-z coordinate system,when the face edges of the body are parallel to coordinate planes,pxx,pyy,pzz are normal stresses and pxy,pxz,pyz,are shear(剪切）stresses.(续)第七章第七章 Pressure At a given point the sum of the normal stresses on any three ortho

15、gonal(直角的）planes is a constant(a scalar).The pressure P is defined asP=-(pxx+pyy+pzz)/3 This is a general definition of the“pressure”.In the special case of a liquid at rest,pxx=pyy=pzz=-P,this is the hydrostatic pressure.In geology,lithostatic pressure is often estimated by using P=gh where is the

16、density,g is the acceleration of gravity,and h is the depth.But it is not always correct near the surface.第七章第七章 (3)(3)Strain（应变）（应变）tensorDefinition:In general,the relative change in the length or in the shape of an object acted by forces is called as strain.This kind of length or shape changes sho

17、uld be recovered after the forces disappear-Elastic deformation.Linear strain A rod is 50cm long initially in the direction x-axis.When a force or forces are applied to it,its length increases to 50.2cm in the x-direction.The relative change in length is(50.2-50)/50.第七章第七章 To measure the relative ch

18、ange,we define u as the length change in x,x as the original length of the rod.On any point inner the rod,the linear strain can be defined asexx=u/x u/x f f f f第七章第七章 In XYZ-coordinate system,in same way we can obtain eyy=v/y,ezz=w/zHere exx,eyy,ezz are normal strains.shear strain Y f X第七章第七章 Suppos

19、e that the graph shown is as a result of external forces,the cross section of the body is deformed to the rhombus(菱形)shown by dashed lines,and in the procession all points move parallel to the x-axis.The area of the cross section has not changed,but the shape has.The angle is a measure of this disto

20、rtion,called shear strain.Here tan=u/y At the limit while y0 =u/y Consider the variation at another direction =v/x第七章第七章 The shear strain in the x-y plane is defined as exy=(+)/2=(u/y+v/x)/2 For three dimension orthogonal coordinates,we also have eyz=(v/z+w/y)/2 exz=(u/z+w/x)/2 Their symmetry gives

21、exy=eyx,exz=ezx,eyz=ezy.第七章第七章Dilatation(体膨胀体膨胀)The sum of normal strains is defined as dilatation =exx+eyy+ezzThe dilatation is a measure of the relative change in volume.For a homogeneous bulk applied by external forces,the relative change in volume is V/(xyz).V=x(1+exx)y(1+eyy)z(1+ezz)-xyzHence V

22、/(xyz)=exx+eyy+ezz+exxezz+eyyezz+exx eyy +exxeyyezz exx+eyy+ezz第七章第七章 (4)Elastic modulus and equations Suppose a body is homogeneous and isotropic,i.e.its properties are independent of both spatial coordinates and directions.Hookes law tell us the stresses in the body are linear combinations of the

23、strains.For instance,pxy=aexx+beyy+cezz+dexy+fexz+geyz (a,b,c,d,f,g are constants)According to elasticity theory,we have pxy=2exy,pxz=2exz,pyz=2eyz where is the modulus of rigidity or shear modulus.第七章第七章 The shear stresses are proportional to the shear strains.On the other hands,the relations of th

24、e normal stresses and normal strains are,e.g.,pxx=+2exxwhere is another elastic modulus.and are called Lam elastic constants.They are difficult to measure directly.For this reason,they are often computed from other elastic parameters.第七章第七章 All the relations to describing stresses and strains can be

25、 written inpij=ij+2eijwhere ij is Kronecker delta-a function.i and j represent x,y or z.when i=j the value of ij is 1,otherwise,0.第七章第七章Other elastic modulassYoungs modulus,E Youngs modulus measures the resistance to extension in a direction.It is defined asE=pxx/exx (if the force only applied in x-

26、direction)Since pxx=+2exxpyy=+2eyy=0pzz=+2ezz=0 第七章第七章To add them Pxx=3+2=(3+2)By symmetry,eyy=ezz=exx+2 eyy,+2eyy=0soE=(3+2)/(+)第七章第七章Incompressibility,This parameter measures the resistance to a change in volume under pressure.It also called bulk modulus.=-dP/dsince P=-(pxx+pyy+pzz)/3 =-(3+2)/3 =-

27、(3+2)/3thus=+2 /3 第七章第七章Poissons ratio,is the ratio of the lateral contraction to the longitudinal extension.Suppose only normal stress pxx acts on the body.Thus=-eyy/exx or =-ezz/exxBecause pyy=+2eyy=(exx+eyy+ezz)+2eyy=0By symmetry,eyy=ezz,and(exx+2eyy)+2eyy=0Hence=/(2(+)第七章第七章 In the earth,Poisson

28、s ratio ranges from 0.1 to 0.38 near the surface.At hydrostatic pressure equivalent to a depth of 13km it ranges from 0.23 to 0.31,except for quartzite=0.15.In the absence of any other information,it is often assumed that=0.25.Other relations =E/(1+)/(1-2)=E/(2(1+)=E/(3(1-)/=2/(1-2)第七章第七章 Wave veloc

29、ities In a infinite,homogeneous,isotropic and elastic medium,only two kinds of waves can propagate,P-wave and S-wave.Their travel velocities Vp and Vs are given respectively by spVV2第七章第七章 Where is the density of the medium.The term P-wave means“primary wave or pressure wave,”since it arrives first

30、or it is caused by pressure.S-wave stands for“secondary wave or shear wave,”because it travel slower than P-wave or it is generated by shear strain.From the equations above,if increases the velocity Vp and Vs should decrease.In fact,it is not real in most cases.Generally the heavier materials have t

31、he higher velocities than the lighter do,because the and increase faster than .For the materials in the earth,assume=0.25,thus=.So we have Vp=1.73Vs第七章第七章 在均匀各向同性介质中，质点的运动方程为在均匀各向同性介质中，质点的运动方程为在无体应力（在无体应力（|f|=0）的情况下，上式变为）的情况下，上式变为 从场论可知，任何一个场从场论可知，任何一个场，均有，均有且有且有则则222222sVtft222)()(222)()(t210,02112

32、212)2(t22222t第七章第七章 根据根据有有 讨论：讨论：1)由于速度是位移对时间的偏导数。因此纵波和横波的速度满足波由于速度是位移对时间的偏导数。因此纵波和横波的速度满足波动方程；动方程；2)由于无旋位移场的散度是无旋应变，无散位移场的旋度是无散应由于无旋位移场的散度是无旋应变，无散位移场的旋度是无散应变。因此无旋应变或正应变与无散应变或切应变均满足波动方程；变。因此无旋应变或正应变与无散应变或切应变均满足波动方程；3)由于无旋场可用标量位来表示，无散场可以用矢量位来表示，并由于无旋场可用标量位来表示，无散场可以用矢量位来表示，并分别设分别设 为标量位和为标量位和 为为矢量位，即有矢

33、量位，即有(续)222222sVt122212pVt222222sVt2222pVt2222sVtspVV,2第七章第七章7.2 地震波及其特征地震波及其特征 (1)(1)纵波纵波 (2)(2)横波横波第七章第七章 (3)(3)瑞雷波瑞雷波 (4)(4)勒夫波勒夫波第七章第七章 (5)(5)SH-wave and SV-wave For the anisotropic media,S-waves may decomposed into two components SH-and SV-waves in their propagation.At the boundaries between di

34、fferent media with the differences of elastic properties,S-wave can generate polarization,i.e.the particles are restricted in a special plane.SH-and SV-wave have slight difference in velocity.第七章第七章 Seismic waves are mechanical waves.They behave the kinds of particle vibration.Since the earthquakes

35、have limit energy so that they only last rather limit time like pulses.We often say that seismic waves signals are wavelets and their periods are irregular.第七章第七章 By applying Fourier analysis we can decompose the wavelets in sine and cosine sequences.The seismic wavelets can be considered the sum of

36、 components of the simple waves with regular periods and various amplitudes.Therefore,we generally use a sine or cosine wave to discuss the feature of wave.第七章第七章 A sine wave can be considered in two ways:at one point,they are periodic in time;at one instant,they are periodic in space.Suppose the mo

37、tion of the particles is along the y-axis,we have y=A sin2(x/-t/T)where T is the period,A is the amplitude of particle vibration,and is wave length,the distance at one instant between two crests or troughs,or between any two adjacent points having same phase.We usually use the terms wave number and

38、frequency.The wave number is k=1/and the frequency is f=1/T.Form the above,we know that the wave velocity can be obtained by V=/T.y=A sin2 k(x-Vt)第七章第七章 Assume that the wave propagates only in x-direction and the particles move only in y-z plane.This means that all particles move in phase-they form

39、wave fronts.We say the waves are plane waves.The movement of the particles may be described by f(x,t)=A sin2k(x-Vt)第七章第七章 7.2.3 7.2.3 球面波球面波For a point source,waves propagate in all directions.If particles move in phase and they constitute spherical wave fronts.We say they are spherical wave.Practic

40、ally,for the far away point sources the spherical wave can be approximately considered as the plane waves.f(r,t)=A sin2k(r-Vt)第七章第七章Dispersion of body waves From a point source the P-and S-waves spread radially from the source along a straight line.The spherical wave fronts still dilate,so that the

41、energy of vibrating particles on the spheres decreases continually.Assume E is the energy of a seismic source,r is the radial distant from the source to a sphere.At one instant the energy per unit area of the sphere E can be written in E=E/(4r2)At any point out of source,the energy is proportional t

42、o the inverse of the square of r and the amplitude is proportional to the inverse of r.They both decay with the increasing of the distance from the source.第七章第七章Absorption and attenuation of body waves Up to now,we have assumed that rock or other materials are perfectly elastic.In fact,pure elastic

43、material does not exist.The energy of the waves transform to heat due to the friction of vibrating particles.The energy and the amplitudes of the waves decay with the traveled distance and the frequency of the wave.A=A0 e-fr第七章第七章7.3 地震体波的传播地震体波的传播研究地震波传播通常有两种途径，一个是依据波动方程的动力学理研究地震波传播通常有两种途径，一个是依据波动方

44、程的动力学理论，另一个是依据地震波走时的射线理论。论，另一个是依据地震波走时的射线理论。(1)(1)费玛原理费玛原理 射线理论的基础是费马原理。费马原理指出：在连续介质中，扰动射线理论的基础是费马原理。费马原理指出：在连续介质中，扰动沿着一条走时稳定的路径传播。若以沿着一条走时稳定的路径传播。若以 t 表示扰动从表示扰动从P点沿着一条路径传到点沿着一条路径传到Q点所用的时间，以点所用的时间，以v(x,y,z)表示扰动的传播速度，以表示扰动的传播速度，以l表示该路径的弧长表示该路径的弧长，则费马原理可以表示为，则费马原理可以表示为稳定值QPzyxvdlt),(第七章第七章(续)换句话说，扰动沿任

45、一射线换句话说，扰动沿任一射线S S传播所用时间传播所用时间t t与沿其它路径传播用时与沿其它路径传播用时一样，即一样，即显然，在均匀介质中，射线为直线，上式可写为显然，在均匀介质中，射线为直线，上式可写为而在非均匀介质中，射线方程的积分形式可写成而在非均匀介质中，射线方程的积分形式可写成VSt Szyxvdst),(0),(Szyxvdst第七章第七章 (2)(2)Snell定律定律 在均匀介质中，地震波射线是直线，在连续介质则为曲线。在非均在均匀介质中，地震波射线是直线，在连续介质则为曲线。在非均匀介质中，当射线到达速度的不连续界面时，其方向会发生偏折，在界匀介质中，当射线到达速度的不连续

46、界面时，其方向会发生偏折，在界面上出现反射波、折射波和转换波。面上出现反射波、折射波和转换波。Snell定律指出了入射与反射和透射定律指出了入射与反射和透射射线之间的关系。射线之间的关系。Snell定律是费马原理的延伸。定律是费马原理的延伸。P1 P1P1P1S1P1P2P1S2第七章第七章(续)如果介质中有界面存在，界面两边的弹性参数及密度各不相同。因如果介质中有界面存在，界面两边的弹性参数及密度各不相同。因为界面两边介质的弹常数和密度都不相同，所以弹性波的速度也不相同为界面两边介质的弹常数和密度都不相同，所以弹性波的速度也不相同。一部分弹性被能量穿过界面，产生透射；另一部分弹性被能量由界面

47、。一部分弹性被能量穿过界面，产生透射；另一部分弹性被能量由界面反射回来。纵波经过界面时产生纵波反射与透射，还可以转换成横被的反射回来。纵波经过界面时产生纵波反射与透射，还可以转换成横被的反射与透射。反射与透射。假设界面是一个平面，当一个单纯的纵波假设界面是一个平面，当一个单纯的纵波P1入射到界面时，便有四入射到界面时，便有四个不同的波同时产生，个不同的波同时产生，P1 P1和和P1 S1表示反射的纵波和横波（表示反射的纵波和横波（SV），），P1P2和和P1S2表示透射纵波和横波（表示透射纵波和横波（SV）。）。横波的质点运动可有两个方向，质点运动与界面垂直的称为横波的质点运动可有两个方向，质

48、点运动与界面垂直的称为SV波，波，质点运动与界面平行的称为质点运动与界面平行的称为SH波。入射的波。入射的SV波在界面上同样可以产生上波在界面上同样可以产生上述四种波，而述四种波，而SH波因质点运动在与界面平行的面上，所以没有纵波产生波因质点运动在与界面平行的面上，所以没有纵波产生。第七章第七章 (1)(1)匀速层状介质中体波的走时匀速层状介质中体波的走时 单个水平界面单个水平界面 在距离振动源不同的地点设置观在距离振动源不同的地点设置观测仪器，接收某种地震波到达的时刻测仪器，接收某种地震波到达的时刻，以距离，以距离x为横轴，到达时刻为横轴，到达时刻t为纵轴为纵轴，所得的曲线称为走时曲线，所得

49、的曲线称为走时曲线(或称时距或称时距曲线曲线)。xcicicABCC C h第七章第七章 假定振动源位于地面假定振动源位于地面A点，地下存在一个水平界面，其深度为点，地下存在一个水平界面，其深度为h，在，在地面地面B点接受到的直达波、反射波和折射走时可分别写成点接受到的直达波、反射波和折射走时可分别写成直达波直达波 反射波反射波 折射波折射波11VxVABt2211141xhVVCBVACt21121cos2VxVihVBCVCCVACtcicicABCC C h第七章第七章多层水平界面反射波多层水平界面反射波 设设Vk,hk,为第为第k层的速度，则有地震波向下传播的射线走时和距离分层的速度，

50、则有地震波向下传播的射线走时和距离分别为别为根据根据Snell定律，有定律，有则则nkkkkiVht1cos2PViViVinn sinsinsin2211nkkkknkkkkPVhPVxPVVht1212)(12)(12xo第七章第七章(2)(2)垂向连续介质（横向均匀）垂向连续介质（横向均匀）若若 hk很小很小,n很大，则有地震波向下传播的射线走时和距离分别为很大，则有地震波向下传播的射线走时和距离分别为由此可见，在连续介质中，射线为一条曲线。由此可见，在连续介质中，射线为一条曲线。hhdzzPVzPVxdzzPVzVt0202)(1)(2)(1)(12xoh第七章第七章根据曲率的定义，根